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On American VIX options under the generalized 3/2 and 1/2 models

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  • Jérôme Detemple
  • Yerkin Kitapbayev

Abstract

In this paper, we extend the 3/2 model for VIX studied by Goard and Mazur and introduce the generalized 3/2 and 1/2 classes of volatility processes. Under these models, we study the pricing of European and American VIX options, and for the latter, we obtain an early exercise premium representation using a free†boundary approach and local time†space calculus. The optimal exercise boundary for the volatility is obtained as the unique solution to an integral equation of Volterra type. We also consider a model mixing these two classes and formulate the corresponding optimal stopping problem in terms of the observed factor process. The price of an American VIX call is then represented by an early exercise premium formula. We show the existence of a pair of optimal exercise boundaries for the factor process and characterize them as the unique solution to a system of integral equations.

Suggested Citation

  • Jérôme Detemple & Yerkin Kitapbayev, 2018. "On American VIX options under the generalized 3/2 and 1/2 models," Mathematical Finance, Wiley Blackwell, vol. 28(2), pages 550-581, April.
    • Handle: RePEc:bla:mathfi:v:28:y:2018:i:2:p:550-581
    DOI: 10.1111/mafi.12153
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    References listed on IDEAS

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    1. Robert Elliott & Tak Kuen Siu & Leunglung Chan, 2007. "Pricing Volatility Swaps Under Heston's Stochastic Volatility Model with Regime Switching," Applied Mathematical Finance, Taylor & Francis Journals, vol. 14(1), pages 41-62.
    2. Jérôme Detemple & Carlton Osakwe, 2000. "The Valuation of Volatility Options," Review of Finance, European Finance Association, vol. 4(1), pages 21-50.
    3. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    4. Grunbichler, Andreas & Longstaff, Francis A., 1996. "Valuing futures and options on volatility," Journal of Banking & Finance, Elsevier, vol. 20(6), pages 985-1001, July.
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    Cited by:

    1. Alessandro Gnoatto & Martino Grasselli & Eckhard Platen, 2022. "Calibration to FX triangles of the 4/2 model under the benchmark approach," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 45(1), pages 1-34, June.
    2. Julia Jiang & Weidong Tian, 2019. "Semi-nonparametric approximation and index options," Annals of Finance, Springer, vol. 15(4), pages 563-600, December.
    3. Hsuan-Ku Liu, 2021. "Perpetual callable American volatility options in a mean-reverting volatility model," Papers 2104.01127, arXiv.org.
    4. Takuji Arai, 2019. "Pricing And Hedging Of Vix Options For Barndorff-Nielsen And Shephard Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(08), pages 1-26, December.
    5. Li Chen & Guang Zhang, 2021. "Hermite Polynomial-based Valuation of American Options with General Jump-Diffusion Processes," Papers 2104.11870, arXiv.org.
    6. Xiaoyu Tan & Chengxiang Wang & Wei Lin & Jin E. Zhang & Shenghong Li & Xuejun Zhao & Zili Zhang, 2021. "The term structure of the VXX option smirk: Pricing VXX option with a two‐factor model and asymmetry jumps," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 41(4), pages 439-457, April.
    7. Takuji Arai, 2019. "Pricing and hedging of VIX options for Barndorff-Nielsen and Shephard models," Papers 1904.12260, arXiv.org.

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